A model for determining the optimal base stock level when the lead time has a change of distribution property

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1 A model for determining the optimal base stock level when the lead time has a change of distribution property R.Jagatheesan 1, S.Sachithanantham 2 1 Research scholar, ManonmaniamSundaranar University, Tirunelveli 12, Tamil Nadu, India. 2 Arignar Anna Government Arts College, Viluppuram, Tamil Nadu, India, rjm.maths@gmail.com 1,sachin7676@yahoo.com 2 AbstractA new type of inventory model using the so called Change of Distribution property is analyzed in this paper. Base Stock system for patient customers is a special type of ordering mechanism in inventory control theory. The inventory process begins with an initial inventory of B units. Whenever a customer s order for r units is received, an inventory replenishment order of r units is placed. Replenishment orders are fulfilled after the lead time L. If the demand exceeds the stock on hand, then assume that the customer will not cancel the order but await the arrival of sufficient stock. Here we assume that L is a random variable and it satisfies the Change of Distribution property and so the distribution undergoes a parametric change after the truncation point. Assuming that the truncation point is a random variable which has the mixed exponential distribution, the optimal value of Base Stock is derived. Index Terms Base Stock, Lead time, Change of Distribution propertyand Truncation point. 1. INTRODUCTION In inventory control theory, one of the important and fame model is Base Stock system for patient customers and these models have been discussed by many authors. Base Stock system for patient customers is an interesting method of ordering mechanism. The inventory initially begins with B number of units. Whenever a customer order for r units is received, an inventory replenishment order for r units is placed immediately. Replenishment orders are filled after the lead time L. The customer s demand is met with as far as possible from the supply on hand. If the total unfilled customer s demand exceeds the supply on hand, then assume that the customer will not cancel the orders but the customers will wait till their requirement is fulfilled. The sum of inventory on hand and order placed is constant in time and equal to B, called as Base Stock. The detailed study of the Base Stock system for patient customers has been discussed by Gaver 1959). The very basic model for base stock system has been discussed by Hanssman 1962). In the sense that after the truncation point the lead time takes parametric change. The change point known as the truncation point and is also assumed to be a random variable. The very basic concept of Change of Distribution property was discussed by Stangl 1995). Ramanarayanan et.al 1998) have discussed the model in which the lead time was assumed to be a random variable. Suresh Kumar 2006) has used this concept in the stock model approach. Sachithanantham et. al 2008) have discussed the modified version of the Base Stock model in which the lead time is assumed to be a random variable and which satisfies the so called Change of distribution property. Henry et. al 2011) have discussed the base stock system for patient customers with the assumptions that the lead time random variable is continuous and it undergoes the Change of distribution property with the assumption that the truncation point itself a random variable, which follows exponential with parameter. The Change of distribution property refers that a random variable takes different probability function after the certain point known as truncation point. In this paper, it is assumed that the lead time random variable has Change of distribution property and the change point itself a random variable, which has the mixed exponential distribution. Under this assumption, the optimal Base Stock level is derived. 2. NOTATIONS B : The Base Stock level. L : A random variable denoting the lead time with the pdf is k.). U : Random variable denoting the interarrival times between successive demands during the lead time with Pdf g.) and cdf G.). : Parameter of inter arrival time distribution. 2019

2 G n.) : The n th convolution of G.). X i : A random variable denoting the magnitude of demand at the i th demand epoch with pdf f.) and F.) is the cdf and X i ~ exp, i = 1,2,3,..n. µ : Parameter of demand distribution. F k.) : The k th convolution of F.). h : Inventory holding cost / unit / time. d : Shortage cost / unit / time. 3. PROBLEM FORMATION Consider the Base Stock system, in which let X be the amount of demand during the lead time L. If fx) is the probability density function of demand, then the Expected Cost per unit time of overages and shortages attributed to the inventory on ground is, Therefore the optimal Base Stock, can be obtained by using. It follows that F 1) Here Fx) is distribution of fx). If there are N demand epochs during L and N is a random variable then the probability that there are exactly n demand during L is [ ] [from the renewal theory arguments] Let X be the total demand during L, say X = X 1 X 2..X N, then the total demand is at the most x, during L, is given by [ ] [ ] Therefore, the expected cost is, [ ] Thus the optimal Base Stock is given by F = [ ] Hence, [ ] 2) In the above, we assume that the lead time L be a continuous random variable with probability density function K.). For the sake of convenience, take L= Y. Therefore the optimal Base Stock is given by [ ] 3) 2020

3 4. DEFINITIONS AND ASSUMPTIONS 4.1. Definition Setting the Clock Back to Zero property SCBZ property) The special property known as Setting the Clock Back to Zero property SCBZ property)is due torajarao. B and Talwaker [4].The family of life distributions {, 0, ϵ Ω } is said to have be SCBZ property is the form of remains unchanged except for value of the parameter. i.e., where ϵ Ω Under the following three operations i) Truncating original distribution of some point 0 ii) Considering the observable distribution for life time and iii) Changing the origin by means of the transformation given by =, 0, where is a Truncation point Assumptions 1. The lead time is continuous random variable and its probability density function undergoes change of distribution property after the truncation point. Here the pdf of lead time random variable is x if X X fx { x if X > X Where is a Truncation point. 2. The truncation point is a random variable, which follows Uniform distribution Hence, f f > Here 3. The interarrival times between successive demand epochs are i.i.d random variables and are assumed to be followed as exponential with parameter 5. MAIN RESULTS In this paper, an improvised model for base stock system for patient customers is discussed with the assumption that the lead time random variable has Change of distribution. It is also assumed that the truncation point itself a random variable, which has Uniform distribution with parameter and. Based on these assumptions the optimal level of base stock system is derived and the variations in the optimal level of base stock with reference to various parameters like holding cost, shortage cost, parameter of inter arrival time distribution and the parameter of demand distribution is studied with the help of numerical illustrations., [ ] 4), > here, 2021

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7 B International Journal of Research in Advent Technology, Vol.6, No.8, August 2018 [ ] ) ), ) ), ) For the fixed values of h, d,, and n, the optimal value of B, can be obtained. 6. NUMERICAL ILLUSTRATIONS The variations in the values of optimal,consequent on the changes in the parametersh, d, and µ have been studied by taking numerical illustrations. The tables and the corresponding graphs are given Casei) For the fixed values of d=3, =2, β=6, =4, = 4, =1, µ=1.5, and n = 5, the optimal value of B is obtained when the variations of h. Table1.The variations in for the changes in the value of h. h h B Fig. 1.The variations in for the changes in the value of h 2025

8 B B B International Journal of Research in Advent Technology, Vol.6, No.8, August Caseii) For the fixed values of h = 5, =2, β=6, =4, = 4, =1, µ=1.5, and n = 5, the optimal value of B is obtained when the variations of d. Table 2. The variations in for the changes in the value ofd. d B Table 4. The variations in for the changes in the value of B Caseiii) For the fixed values of h = 5, d= 3, β=6, =4, = 4, =1, µ=1.5, and n = 5, the optimal value of B is obtained when the variations of µ Table 3. The variations in for the changes in the value of B Fig.4.The variations in for the changes in the value of. 7. FINDINGS AND CONCLUSIONS From the tables and graphs, it is observed that, As the holding cost h) increases, the optimal Base Stock decreases. Hence it suggests less base stock Caseiv) For the fixed values of h = 5, d= 3, β=6, =4, = 4, =1, and n = 5, the optimal value of B is obtained when the variations of Fig. 2.The variations in for the changes in the value of d. d Fig. 3. The variations in for the changes in the value of. λ 2026

9 As the shortage cost d) increases, the optimal Base Stock increases and it tells us to maintain the more base stock level. As, the parameter of inter arrival time distribution between demand epochs increases, the optimal Base Stock decreases. It is inferred that the total demand is distributed at different demand epochs with lesser demand and it suggest lesser base stock level. As µ), the parameter of demand distribution increases, the optimal Base Stock decreases. It is inferred that the average demand is distributed at different demand epochs with lesser demand and it suggest lesser base stock level. REFFERENCES [1] D.P.,Jr.Gaver. On Base Stock Level Inventory Control. Operations Res. 7: , [2] F. Hanssmann. Operations Research in Production and Inventory Control. John Wiley and sons, inc. New York, [3] B. Raja Rao. Life Expectancy for a class of life distributions having the setting the Clock back to Zero property. Mathematical BioSciences, 98, pp , [4] R. Ramanarayanan,V. Ramachandran and R. Sathiyamoorthi, Base Stock system for patient customers when inter arrival times of demands are dependent. ASR. Vol. 2. No. 1, pp 16, [5] D.K. Stangl. Exponential change point model, statistics in Medicine, Vol.14, pp , [6] R. Suresh Kumar. Shock Model when the threshold has a change of Distribution after a change point. Journal of Indian Acad Math, Vol.28,No.1):pp.7384, [7] S. Sachithanantham, V.Ganesan. and R. Sathiyamoorthi. A model of Base Stock system for patient customers with lead time distribution undergoing a parametric change Journal of Ultra scientist of physical sciences. Vol.20No.3)M, pp , [8] L. Henry, C. Selvaraj and S. Sachithanantham. A Model for Base stock system for patient customers with lead time having a change of distribution, Journal of Ultra scientist of physical sciences. Vol.23No.3A), pp ,